\(\int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx\) [411]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\frac {2^{-3-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}+\frac {2^{-3-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}+\text {Int}\left (x^m \coth (a+b x),x\right ) \]

[Out]

2^(-3-m)*exp(2*a)*x^m*GAMMA(1+m,-2*b*x)/b/((-b*x)^m)+2^(-3-m)*x^m*GAMMA(1+m,2*b*x)/b/exp(2*a)/((b*x)^m)+Uninte
grable(x^m*coth(b*x+a),x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx \]

[In]

Int[x^m*Cosh[a + b*x]^2*Coth[a + b*x],x]

[Out]

(2^(-3 - m)*E^(2*a)*x^m*Gamma[1 + m, -2*b*x])/(b*(-(b*x))^m) + (2^(-3 - m)*x^m*Gamma[1 + m, 2*b*x])/(b*E^(2*a)
*(b*x)^m) + Defer[Int][x^m*Coth[a + b*x], x]

Rubi steps \begin{align*} \text {integral}& = \int x^m \coth (a+b x) \, dx+\int x^m \cosh (a+b x) \sinh (a+b x) \, dx \\ & = \int x^m \coth (a+b x) \, dx+\int \frac {1}{2} x^m \sinh (2 a+2 b x) \, dx \\ & = \frac {1}{2} \int x^m \sinh (2 a+2 b x) \, dx+\int x^m \coth (a+b x) \, dx \\ & = \frac {1}{4} \int e^{-i (2 i a+2 i b x)} x^m \, dx-\frac {1}{4} \int e^{i (2 i a+2 i b x)} x^m \, dx+\int x^m \coth (a+b x) \, dx \\ & = \frac {2^{-3-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}+\frac {2^{-3-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}+\int x^m \coth (a+b x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 16.83 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx \]

[In]

Integrate[x^m*Cosh[a + b*x]^2*Coth[a + b*x],x]

[Out]

Integrate[x^m*Cosh[a + b*x]^2*Coth[a + b*x], x]

Maple [N/A] (verified)

Not integrable

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

\[\int x^{m} \cosh \left (b x +a \right )^{3} \operatorname {csch}\left (b x +a \right )d x\]

[In]

int(x^m*cosh(b*x+a)^3*csch(b*x+a),x)

[Out]

int(x^m*cosh(b*x+a)^3*csch(b*x+a),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right ) \,d x } \]

[In]

integrate(x^m*cosh(b*x+a)^3*csch(b*x+a),x, algorithm="fricas")

[Out]

integral(x^m*cosh(b*x + a)^3*csch(b*x + a), x)

Sympy [F(-1)]

Timed out. \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\text {Timed out} \]

[In]

integrate(x**m*cosh(b*x+a)**3*csch(b*x+a),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right ) \,d x } \]

[In]

integrate(x^m*cosh(b*x+a)^3*csch(b*x+a),x, algorithm="maxima")

[Out]

integrate(x^m*cosh(b*x + a)^3*csch(b*x + a), x)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right ) \,d x } \]

[In]

integrate(x^m*cosh(b*x+a)^3*csch(b*x+a),x, algorithm="giac")

[Out]

integrate(x^m*cosh(b*x + a)^3*csch(b*x + a), x)

Mupad [N/A]

Not integrable

Time = 2.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\int \frac {x^m\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]

[In]

int((x^m*cosh(a + b*x)^3)/sinh(a + b*x),x)

[Out]

int((x^m*cosh(a + b*x)^3)/sinh(a + b*x), x)